📐 The Linear Demand Function
Standard form: The linear demand function is written as: $$Q_d = a - bP$$ where a = quantity demanded when price is zero (the Q-intercept) and b = the responsiveness of quantity demanded to a change in price.
- a is the autonomous demand — the quantity demanded at P = 0 (the intercept on the Q-axis).
- b is the slope parameter — it tells you how many units quantity demanded falls for each $1 increase in price.
- P is the price of the good.
- The negative sign reflects the inverse relationship: as P rises, Qd falls (law of demand).
Finding the P-intercept
The P-intercept (where the demand curve hits the vertical axis) is found by setting $Q_d = 0$: $$0 = a - bP \implies P = \frac{a}{b}$$ This is the choke price.
Worked example: If $Q_d = 100 - 5P$, then: a = 100 (Q-intercept), b = 5, P-intercept = 100/5 = $20. At $20, nobody buys. At $0, demand is 100 units.
📈 Graphing and Shifts of the Linear Demand Curve
Graphing convention in economics
Axes are swapped!: In economics, we plot P on the vertical axis and Q on the horizontal axis — the opposite of the mathematical convention. So $Q_d = a - bP$ becomes $P = \frac{a}{b} - \frac{1}{b}Q_d$ when expressed with P as the subject. The slope on the diagram is $-\frac{1}{b}$, NOT $-b$.
Shifts vs movements
- A change in P causes a movement ALONG the demand curve (change in Qd).
- A change in a causes a SHIFT of the entire demand curve (change in demand).
- If a increases → demand shifts RIGHT (more demanded at every price).
- If a decreases → demand shifts LEFT (less demanded at every price).
- A change in b changes the SLOPE (steepness) of the curve.
Non-price determinants (income, tastes, population, substitutes, complements) change the value of a. A steeper curve (smaller b) means consumers are less responsive to price — more inelastic.